Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$-plane.
A) $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$
B) $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$
C) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$
D) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$
E) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$

Question

Quizplus · Accepted Answer

Since the sphere touches the $xy$-plane, the $z$-coordinate of its center must be the radius of the sphere. Therefore, the center is $(7,-5,7)$ with a radius of $7$.

Using the equation of a sphere with center $(h,k,l)$ and radius $r$: $$(x-h)^2+(y-k)^2+(z-l)^2=r^2$$ we can plug in the values to get the equation of the sphere: $$(x-7)^2+(y+5)^2+(z-7)^2=7^2$$ which simplifies to $$(x-7)^2+(y+5)^2+(z-7)^2=49$$ which matches choice D.

Find an Equation of the Sphere with Center (7,−5,7)( 7 , - 5,7 )(7,−5,7)

Find an Equation of the Sphere with Center $( 7 , - 5,7 )$